Log Growth

When nutrients are in excess, microbial growth is described well by the equation:
dX/dt = µ X
where X is concentration of organisms, t is time, and the specific growth rate coefficient is µ .
This differential equation can be integrated within limits to give:
X = Xo EXP (µ * t)

You can solve this with the following Java applet:

Instructions for solving with Maple

  1. At R.P.I. using our Unix system, click on the menu selection to get a Maple window.
  2. Type the following exactly:
    x(t):=.5*exp(t);
    plot (x(t),t=0..5);
  3. Position the cursor just at the end of the first line you typed and strike the enter key.
  4. Strike the enter key again and wait for a few seconds until a graph of X vs. t appears.
  5. Experiment by using the cursor and keyboard to change one of the numbers in either line that you typed and repeating Instruction 3.

DISCUSSION

When you changed the coefficient in the 1st equation, the graph did not seem to change. The main difference was in the numbers on the ordinate. Maple autoscales its graphs, and this can give almost the same shape for a graph.

Changing the numbers in the plot line has a much more noticable change. If you specify a large interval for the time axis, the plot may blow up because the concentration of organisms goes beyong the number range for your computer.

Unlimited growth makes no sense. Nutrients would be exhausted and growth would stop. See the Monod equation and the comments about growth-limiting nutrient.